| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2002 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Probability Distributions |
| Type | Two unknowns from sum and expectation |
| Difficulty | Moderate -0.8 This is a straightforward S1 question testing basic probability distribution properties. Part (a) requires setting up E(X) = Σxp(x) and solving a simple linear equation. Part (b) uses the standard variance formula with values already found. Part (c) is trivial since X only takes values 0,1,2. All parts are routine textbook exercises with no problem-solving insight required, making it easier than average. |
| Spec | 5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables |
| \(x\) | 0 | 1 | 2 |
| P(\(X = x\)) | \(\frac{1}{3}\) | \(a\) | \(\frac{2}{3} - a\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(a + 2(\frac{3}{2} - a) = \frac{5}{6}\) | M1 | Use of \(E(X)\) correct equation |
| \(a = \frac{1}{3}\) | A1 | |
| A1 (3) | Allow \(\frac{1}{3}\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\text{Var}(X) = [\frac{1}{3} \cdot \frac{1}{2} + 2 \cdot \frac{1}{4} - (\frac{2}{3})^2]\) | M1 | Use of \(\sum x^2 P(x+x) - \mu^2\) |
| \(= \frac{17}{36} = 0.472\) | M1 | Allow \(\frac{17}{36}\) |
| A1 (3) |
| Answer | Marks |
|---|---|
| \(P(X \leq 1.5) = P(0) + P(1) = \frac{2}{3} + \frac{1}{2} = \frac{7}{6}\) | B1(0) |
## Part (a)
$a + 2(\frac{3}{2} - a) = \frac{5}{6}$ | M1 | Use of $E(X)$ correct equation
$a = \frac{1}{3}$ | A1 |
| A1 (3) | Allow $\frac{1}{3}$
## Part (b)
$\text{Var}(X) = [\frac{1}{3} \cdot \frac{1}{2} + 2 \cdot \frac{1}{4} - (\frac{2}{3})^2]$ | M1 | Use of $\sum x^2 P(x+x) - \mu^2$
$= \frac{17}{36} = 0.472$ | M1 | Allow $\frac{17}{36}$
| A1 (3) |
## Part (c)
$P(X \leq 1.5) = P(0) + P(1) = \frac{2}{3} + \frac{1}{2} = \frac{7}{6}$ | B1(0) |
---A discrete random variable $X$ has the probability function shown in the table below.
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
$x$ & 0 & 1 & 2 \\
\hline
P($X = x$) & $\frac{1}{3}$ & $a$ & $\frac{2}{3} - a$ \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Given that E($X$) = $\frac{2}{3}$, find $a$. [3]
\item Find the exact value of Var ($X$). [3]
\item Find the exact value of P($X \leq 15$). [1]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2002 Q3 [7]}}